A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

Sphere

Time a body takes to roll down an inclined plane is : t = \(\sqrt{\frac{2l(1 + \frac{k^2}{r^2})}{g \sin {\theta}}}\)

\(\frac{t_{disk}}{t_{sphere}} = \sqrt{\frac{1 + \frac{k_d^2}{R^2}}{1 + \frac{k_s^2}{R^2}}}\)

\(k_d = \sqrt{\frac{1}{2}}R ; k_s = \sqrt{\frac{2}{5}}R \)

\(\frac{t_{disk}}{t_{sphere}} = \sqrt{\frac{1 + \frac{1}{2R^2}}{1+\frac{2}{5R^2}}}\)

\(\frac{t_{disk}}{t_{sphere}} = \sqrt{\frac{15}{14}}\)

Time taken by sphere is less.